On the Number of Limit Cycles for a Generalization of Liénard Polynomial Differential Systems

Abstract

We study the number of limit cycles of the polynomial differential systems of the form [Formula: see text] where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = -x using the averaging theory of third order.

Limit cycles of Liénard polynomial systems type by averaging method

“…• 4) for k = 6, 7, 8, 9 andĤ(5, 6) =Ĥ(6, 5). In 1998 Gasull and Torregrosa [13] obtained upper bounds forĤ (7,6),Ĥ(6, 7),Ĥ(7, 7), and H (4,20). In 2006 Yu and Han in [30] give some accurate values ofĤ(m, n) =Ĥ(n, m) for n = 4, m = 10, 11, 12, 13; n = 5, m = 6, 7, 8, 9; n = 6, m = 5, 6.…”

“…where g 1 (x), f 1 (x), g 2 (x) and f 2 (x) have degree k, l, m and n, respectively. • In 2013, Llibre and Valls studied using the averaging theory of third order the following systems (see [19] and [20] respectively)…”

Limit cycles of generalized Liénard polynomial differential systems via averaging theory

“…where studied in the papers [24], [1] and [25,26], respectively. Moreover, the number of medium amplitude limit cycles bifurcating from the centerẋ…”

On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

“…This problem restricted to continuous planar polynomial differential systems is the well known Hilbert's 16th problem, see for example [Li, 2003]. Up to now, there have been many achievements concerning the existence, uniqueness and the number of limit cycles, see for example [García et al, 2014;Justino & Jorge, 2012;Li & Llibre, 2012;Llibre, 2010;Llibre & Mereu, 2013Llibre et al, 2015;Llibre & Valls, 2012, 2013a, 2013bLloyd & Lynch, 1988;Martins & Mereu, 2014;Shen & Han, 2013;Sun, 1992;Xiong & Zhong, 2013] and references therein. A limit cycle bifurcating from a single degenerate singular point is called a small amplitude limit cycle, and the one bifurcating from periodic orbits of a linear center is called a medium amplitude limit cycle.…”